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Application of cubic B-spline finite element technique to the thermistor problem. (English) Zbl 1038.65057
Summary: A numerical solution to the thermistor problem is obtained using cubic B-spline finite elements. The resulting system of ordinary differential equations is solved by the finite-difference method. Excellent agreement is obtained between the numerical results and the analytic solution for the three phases.

65L05 Numerical methods for initial value problems
80A20 Heat and mass transfer, heat flow (MSC2010)
80M10 Finite element, Galerkin and related methods applied to problems in thermodynamics and heat transfer
80M20 Finite difference methods applied to problems in thermodynamics and heat transfer
65L12 Finite difference and finite volume methods for ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems, general theory
Full Text: DOI
[1] Alexander, M.E.; Morris, J.LI., Galerkin methods for some model equations for nonlinear dispersive waves, J. comput. phys., 30, 428-451, (1979) · Zbl 0407.76014
[2] Caldwell, J.; Smith, P., Solution of burger’s equation with a large Reynolds number, Appl. math. modell., 6, 239-253, (1982)
[3] Jain, P.C.; Holla, D.N., Numerical solution of coupled burgers’ equation, Int. J. nonlinear mech., 13, 213-222, (1978) · Zbl 0388.76049
[4] Lohar, B.L.; Jain, P.C., Variable mesh cubic spline technique for N-wave solution of burgers’ equation, J. comp. phys., 39, 433-442, (1981) · Zbl 0471.76073
[5] Herbst, B.M.; Schoombie, S.W.; Mitchell, A.R., A moving petrov – galerkin method for transport equations, Int. J. numer. meth. engrg., 18, 1321-1336, (1982) · Zbl 0485.65093
[6] Ali, A.H.A.; Gardner, L.R.T.; Gardner, G.A., A collocation solution for burgers’ equation using cubic B-spline finite elements, Comput. meth. appl. mech. engrg., 100, 325-337, (1992) · Zbl 0762.65072
[7] Gardner, L.R.T.; Gardner, G.A., A two-dimensional bi-cubic B-spline finite element: a study of MHD duct flow, Comp. meth. appl. mech. eng., 124, 365-375, (1995) · Zbl 0844.65073
[8] Gardner, L.R.T.; Gardner, G.A.; Dogan, A., A petrov – galerkin finite element scheme for burgers’ equation, Arabian J. sci. engrg., 22, 99-109, (1997) · Zbl 0908.65089
[9] Antonisev, S.N.; Chpot, M., The thermistor problem: existence, smoothness, uniqueness, blowup, SIAM J. math. anal., 25, 4, 1157-1166, (1994)
[10] Cimatti, G., A bound for the temperature in the thermistor problem, Q. appl. maths., 40, 15-22, (1988) · Zbl 0694.35139
[11] Cimatti, G., Remark on existence and uniqueness for the thermistor problem under mixed boundary conditions, Q. appl. maths., 47, 117-121, (1989) · Zbl 0694.35137
[12] A.C. Fowler, S.D. Howison Temperature surges in thermistor, in: J. Manley et al. (Eds.), Proceedings of the 3rd Euro. Conf. Maths. Industry, Kluwer, Stuttgart, 1990, pp. 197-204
[13] Fowler, A.C.; Howison, S.D.; Hinch, E.J., Temperature surges in current-limiting circuit devices, SIAM J. appl. math., 52, 4, 998-1011, (1992) · Zbl 0800.80001
[14] Howison, S.D.; Rodrigues, J.F.; Shillor, M., Stationary solutions to the thermistor problem, J. math. anal. appl., 174, 573-588, (1993) · Zbl 0787.35033
[15] Kutluay, S.; Bahadır, A.R.; Ozdes, A., Various methods to the thermistor problem with a bulk electrical conductivity, Int. J. numer. meth. engng., 45, 1-12, (1999) · Zbl 0941.78011
[16] Kutluay, S.; Bahadır, A.R.; Ozdes, A., A variety of finite difference methods to the thermistor with a new modified electrical conductivity, Appl. maths. comput., 106, 205-213, (1999) · Zbl 1049.80501
[17] Westbrook, D.R., The thermistor: A problem in heat and current flow, Numer. meth. pdes, 5, 259-273, (1989) · Zbl 0676.65128
[18] Zhou, S.; Westbrook, D.R., Numerical solutions of the thermistor equations, J. comput. appl. math., 79, 101-118, (1997) · Zbl 0885.65147
[19] Prenter, P.M., Splines and variational methods, (1975), Wiley New York · Zbl 0344.65044
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